Isomorphic Modules Research Articles

Polynomial-time algorithms for quadratic isomorphism of polynomials: The regular case

Let f=(f1,…,fm) and g=(g1,…,gm) be two sets of m≥1 nonlinear polynomials in K[x1,…,xn] (K being a field). We consider the computational problem of finding–if any–an invertible transformation on the variables mapping f to g. The corresponding equivalence problem is known as Isomorphism of Polynomials with one Secret (IP1S) and is a fundamental problem in multivariate cryptography. Amongst its applications, we can cite Graph Isomorphism (GI) which reduces to equivalence of cubic polynomials with respect to an invertible linear change of variables, according to Agrawal and Saxena. The main result is a randomized polynomial-time algorithm for solving IP1S for quadratic instances–a particular case of importance in cryptography.To this end, we show that IP1S for quadratic polynomials can be reduced to a variant of the classical module isomorphism problem in representation theory. We show that we can essentially linearize the problem by reducing quadratic-IP1S to test the orthogonal simultaneous similarity of symmetric matrices; this latter problem was shown by Chistov, Ivanyos and Karpinski (ISSAC 1997) to be equivalent to finding an invertible matrix in the linear space Kn×n of n×n matrices over K and to compute the square root in a certain representation in a matrix algebra. While computing square roots of matrices can be done efficiently using numerical methods, it seems difficult to control the bit complexity of such methods. However, we present exact and polynomial-time algorithms for computing a representation of the square root of a matrix in Kn×n, for various fields (including finite fields), as a product of two matrices. Each coefficient of these matrices lies in an extension field of K of polynomial degree. We then consider #IP1S, the counting version of IP1S for quadratic instances. In particular, we provide a (complete) characterization of the automorphism group of homogeneous quadratic polynomials. Finally, we also consider the more general Isomorphism of Polynomials (IP) problem where we allow an invertible linear transformation on the variables and on the set of polynomials. A randomized polynomial-time algorithm for solving IP when f=(x1d,…,xnd) is presented. From an algorithmic point of view, the problem boils down to factoring the determinant of a linear matrix (i.e. a matrix whose components are linear polynomials). This extends to IP a result of Kayal obtained for PolyProj.

The module isomorphism problem reconsidered

Algorithms to decide isomorphism of modules have been honed continually over the last 30 years, and their range of applicability has been extended to include modules over a wide range of rings. Highly efficient computer implementations of these algorithms form the bedrock of systems such as GAP and Magma, at least in regard to computations with groups and algebras. By contrast, the fundamental problem of testing for isomorphism between other types of algebraic structures – such as groups, and almost any type of algebra – seems today as intractable as ever. What explains the vastly different complexity status of the module isomorphism problem?This paper argues that the apparent discrepancy is explained by nomenclature. Current algorithms to solve module isomorphism, while efficient and immensely useful, are actually solving a highly constrained version of the problem. We report that module isomorphism in its general form is as hard as algebra isomorphism and graph isomorphism, both well-studied problems that are widely regarded as difficult. On a more positive note, for cyclic rings we describe a polynomial-time algorithm for the general module isomorphism problem. We also report on a Magma implementation of our algorithm.

Open projections in operator algebras I: Comparison theory

We begin a program of generalizing basic elements of the theory of comparison, equivalence, and subequivalence, of elements in C � -algebras, to the setting of more general algebras. In particular, we follow the recent lead of Lin, Ortega, Rordam, and Thiel of studying these equivalences, etc., in terms of open projections or module isomorphisms. We also define and characterize a new class of inner ideals in operator algebras, and develop a matching theory of open partial isometries in operator ideals which simultaneously generalize the open projections in operator algebras (in the sense of the authors and Hay), and the open partial isometries (tripotents) introduced by the authors.

Fraction-free algorithm for the computation of diagonal forms matrices over Ore domains using Gröbner bases

This paper is a sequel to “Computing diagonal form and Jacobson normal form of a matrix using Gröbner bases” (Levandovskyy and Schindelar, 2011). We present a new fraction-free algorithm for the computation of a diagonal form of a matrix over a certain non-commutative Euclidean domain over a computable field with the help of Gröbner bases. This algorithm is formulated in a general constructive framework of non-commutative Ore localizations of G-algebras (OLGAs). We use the splitting of the computation of a normal form for matrices over Ore localizations into the diagonalization and the normalization processes. Both of them can be made fraction-free. For a given matrix M over an OLGA R, we provide a diagonalization algorithm to compute U,V and D with fraction-free entries such that UMV=D holds and D is diagonal. The fraction-free approach allows to obtain more information on the associated system of linear functional equations and its solutions, than the classical setup of an operator algebra with coefficients in rational functions. In particular, one can handle distributional solutions together with, say, meromorphic ones. We investigate Ore localizations of common operator algebras over K[x] and use them in the unimodularity analysis of transformation matrices U,V. In turn, this allows to lift the isomorphism of modules over an OLGA Euclidean domain to a smaller polynomial subring of it. We discuss the relation of this lifting with the solutions of the original system of equations. Moreover, we prove some new results concerning normal forms of matrices over non-simple domains. Our implementation in the computer algebra system Singular:Plural follows the fraction-free strategy and shows impressive performance, compared with methods which directly use fractions. In particular, we experience a moderate swell of coefficients and obtain simple transformation matrices. Thus the method we propose is well suited for solving nontrivial practical problems.

Theta series and function field analogue of Gross formula

Let k = F_q(t) , with q odd. In this article we introduce «definite» (with respect to the infinite place of k ) Shimura curves over k , and establish Hecke module isomorphisms between their Picard groups and the spaces of Drinfeld type «new» forms of corresponding level. An important application is a function field analogue of Gross formula for the central critical values of Rankin type L -series coming from automorphic cusp forms of Drinfeld type.

Homological Obstructions to String Orientations

We observe that the Poincare duality isomorphism for a string manifold is an isomorphism of modules over the subalgebra A(2) of the modulo 2 Steenrod algebra. In particular, the pattern of the operations Sq^1, Sq^2, and Sq^4 on the cohomology of a string manifold has a symmetry around the middle dimension. We characterize this kind of cohomology operation duality in term of the annihilator of the Thom class of the negative tangent bundle, and in terms of the vanishing of top-degree cohomology operations. We also indicate how the existence of such an operation-preserving duality implies the integrality of certain polynomials in the Pontryagin classes of the manifold.

Deterministic Polynomial Time Algorithms for Matrix Completion Problems

We present new deterministic algorithms for several cases of the maximum rank matrix completion problem (for short matrix completion), i.e., the problem of assigning values to the variables in a given symbolic matrix to maximize the resulting matrix rank. Matrix completion is one of the fundamental problems in computational complexity. It has numerous important algorithmic applications, among others, in computing dynamic transitive closures or multicast network codings [N. J. A. Harvey, D. R. Karger, and K. Murota, Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, 2005, pp. 489–498; N. J. A. Harvey, D. R. Karger, and S. Yekhanin, Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, 2006, pp. 1103–1111]. We design efficient deterministic algorithms for common generalizations of the results of Lovász and Geelen on this problem by allowing linear polynomials in the entries of the input matrix such that the submatrices corresponding to each variable have rank one. Our methods are algebraic and quite different from those of Lovász and Geelen. We look at the problem of matrix completion in the more general setting of linear spaces of linear transformations and find a maximum rank element there using a greedy method. Matrix algebras and modules play a crucial role in the algorithm. We show (hardness) results for special instances of matrix completion naturally related to matrix algebras; i.e., in contrast to computing isomorphisms of modules (for which there is a known deterministic polynomial time algorithm), finding a surjective or an injective homomorphism between two given modules is as hard as the general matrix completion problem. The same hardness holds for finding a maximum dimension cyclic submodule (i.e., generated by a single element). For the “dual” task, i.e., finding the minimal number of generators of a given module, we present a deterministic polynomial time algorithm. The proof methods developed in this paper apply to fairly general modules and could also be of independent interest.

On the Coates-Sinnott Conjecture

In [5], Coates and Sinnott formulated a far reaching conjecture linking the values ΘF/k,S (1 — n) for even integers n ≥ 2 of an S -imprimitive, Galois-equivariant L -function ΘF/k,S associated to an abelian extension F/k of totally real number fields to the annihilators over the group ring ℤ[G (F/k)] of the even Quillen K -groups K2n–2 (OF) associated to the ring of integers OF of the top field F. In the same paper, Coates and Sinnott essentially prove the 𝓁 -adic étale cohomological version of their conjecture, in which K2n–2(OF) is replaced by H2 et (OF [1/𝓁 ], ℤ(n)), for all primes 𝓁 > 2, under the hypothesis that k = ℚ. Refinements of this result for k = ℚ, involving Fitting ideals rather than annihilators of H2et (OF [1/𝓁], ℤ𝓁(n)), were obtained in particular cases by Cornacchia–Østvaer [7] and in general by Kurihara [14]. More recently, Burns and Greither [3] proved the same type of refinements (involving Fitting ideals of étale cohomology groups) for arbitrary totally real base fields k, but working under the very strong hypothesis that the Iwasawa μ -invariants μF,𝓁 vanish for all odd primes 𝓁. In this paper, we study a class of abelian extensions of an arbitrary totally real base field k including, for example, subextensions of real cyclotomic extensions of type k (ζ)+/k, where p is an odd prime. For this class of extensions, we prove similar refinements of the étale cohomological version of the Coates–Sinnott conjecture, under no vanishing hypotheses for the Iwasawa μ-invariants in question. Our methods of proof are different from the ones employed in [3], [14] and [7]. We build upon ideas developed by Greither in [10] and Wiles in [23] and [22], in the context of Brumer's Conjecture. If the Quillen–Lichtenbaum Conjecture is proved (and a proof seems tobe within reach), then we have canonical ℤ𝓁[G (F/k)]-module isomorphisms for all n ≥ 2, all i = 1,2, and all primes 𝓁 > 2, and all these results will yield proofs of the original K -theoretic version of the Coates–Sinnott Conjecture, in the cases and under the various hypotheses mentioned above (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

On isometries for convolutional codes

In this paper we will discuss isometries and strong isometries for convolutional codes. Isometries are weight-preserving module isomorphisms whereas strong isometries are, in addition, degree-preserving. Special cases of these maps are certain types of monomial transformations. We will show a form of MacWilliams Equivalence Theorem, that is, each isometry between convolutional codes is given by a monomial transformation.Examples show that strong isometries cannot be characterized this way, but special attention paid to the weight adjacency matrices allows for further descriptions. Various distance parameters appearing in the literature on convolutional codes will be discussed as well.

Isomorphism of Hilbert modules over stably finite C ∗-algebras

It is shown that if A is a stably finite C ∗-algebra and E is a countably generated Hilbert A-module, then E gives rise to a compact element of the Cuntz semigroup if and only if E is algebraically finitely generated and projective. It follows that if E and F are equivalent in the sense of Coward, Elliott and Ivanescu (CEI) and E is algebraically finitely generated and projective, then E and F are isomorphic. In contrast to this, we exhibit two CEI-equivalent Hilbert modules over a stably finite C ∗-algebra that are not isomorphic.

Shoikhet's Conjecture and Duflo Isomorphism on (Co)Invariants

In this paper we prove a conjecture of B. Shoikhet. This conjecture states that the tangent isomorphism on homology, between the Poisson homology associated to a Poisson structure on R d and the Hochschild homology of its quantized star-product algebra, is an isomorphism of modules over the (isomorphic) respective cohomology algebras. As a consequence, we obtain a version of the Duflo isomorphism on coinvariants.

Algebraic representation of mappings between submodule lattices

We show that under certain weak conditions on the module R M, every mapping $$ f:\mathfrak{L}\left( {_R M} \right) \to \mathfrak{L}\left( {_S N} \right) $$ between the submodule lattices which preserves arbitrary joins and “disjointness” has a unique representation of the form f(u) = 〈h[ S B R × R U]〉 for all u ∈ $$ \mathfrak{L}\left( {_R M} \right) $$, where S B R is some bimodule and h is an R-balanced mapping. Furthermore, f is a lattice homomorphism if and only if B R is flat and the induced S-module homomorphism $$ \bar h:_S B \otimes _R M \to _S N $$ is monic. If S N also satisfies the same weak conditions, then f is a lattice isomorphism if and only if B R is a finitely generated projective generator, S ≅ End(B R ) canonically, and $$ \bar h:_S B \otimes _R M \to _S N $$ is an S-module isomorphism, i.e., every lattice isomorphism is induced by a Morita equivalence between R and S and a module isomorphism.

Single elements and module isomorphisms of some operator algebra modules

In this paper, we first introduce the concept of single elements in a module. A systematic study of single elements in the AlgL-module U is initiated, where £ is a completely distributive subspace lattice on a Hilbert space H. Furthermore, as an application of single elements, we study module isomorphisms between norm closed AlgN-modules, where N is a nest, and obtain the following result: Suppose that U, V are norm closed AlgN-modules and that Φ: U → V is a module isomorphism. Then U = V and there exists a non-zero complex number A such that Φ(T) = λT, VT ∈ U.

Equivariant flow equivalence for shifts of finite type, by matrix equivalence over group rings

Let G denote a nite group. In this paper, by a G shift of nite type ( G-SFT) we will mean a shift of nite type (SFT) together with a continuous G-action which commutes with the shift, where in addition the action is free and the SFT is irreducible and is non-trivial (contains more than one orbit). We will classify these systems up to G-5ow equivalence. This equivalence relation can be described in terms of G-SFTs, skew products or suspension 5ows (x 2). For example, two G-SFTs are G-5ow equivalent if and only if there exists an orientation-preserving homeomorphism between their mapping tori which commutes with the induced G actions. A G-SFT can be presented by a nite square matrix A over ZþG, the positive cone of the integral group ring ZG (W. Parry, personal communication 2001). Let ðIAÞ1 denote the NN matrix whose upper left corner is IA and which otherwise equals the innite identity matrix. Let E ðZGÞ be the group of NN matrices generated by basic elementary matrices (those which di;er from I in at most one entry, which must be o;-diagonal) over ZG. Let WðAÞ denote the weight class of A (Denition 4.1): the conjugacy class in G of the group of weights of loops based at a xed vertex. We show that the weight class is an invariant of G-5ow equivalence. When WðAÞ¼WðBÞ¼G, we will show that G-SFTs presented by matrices A and B are G-5ow equivalent if and only if there are matrices U and V in EðZGÞ such that UðIAÞ1V ¼ðIBÞ1 (Theorem 6.1). The complete classication up to G-5ow equivalence, which allows the possibility WðAÞ ( G, has a more complicated statement (Theorem 6.4). In the case that G is trivial, our classication reduces to the familiar classication of Franks ( 14) by cokernel group and determinant. When G is non-trivial, the classication up to E ðZGÞ is much more diBcult and interesting, and remains an open problem. We consider these algebraic issues inxx 8 and 9. In x 8, we give the modest requisite K-theory terminology and background, and for the case G¼ Z=2 we give a constructive partial result (Theorem 8.1) and some very concrete illustrative examples (Examples 8.6 and 8.7) which indicate how the ZG-equivalence problem becomes more diBcult when G is non-trivial (that is, ZG6 Z). In x 9, we consider EðZGÞ-equivalence of injective matrices. In this case, GLðZGÞ-equivalence amounts to isomorphism of cokernel modules, and the renement toE ðZGÞ-equivalence is classied by K 1ðZGÞ=H for an associated subgroup H ofS K 1ðZGÞ. As one consequence, if G is abelian and detðIAÞ is not a zerodiviso r in ZG, then detðIAÞ determines the G-5ow equivalence class up

PROJECTIVE PRESENTATIONS OF FINITELY GENERATED MODULES WITH LARGE ANNIHILATORS

ABSTRACT Let R be an integral domain, and let A be an “almost” finitely presented R -module such that R / Ann ( A ) is a finite direct sum of quasilocal rings. We characterize the relationship between the modules that can occur as kernels of finite rank projective presentations of A, and we give a criterion for when local isomorphism of torsionless modules implies an equivalence of the two modules up to summands of projective R-modules.

Results on undecidability of isomorphism of forms over polynomial rings

We show the undecidability of the question of isomorphism of forms over a polynomial ring $ R[t_1,\ldots,t_n] $, assuming a hypothesis about units in certain quaternion rings. Assuming this, it follows that isomorphisms of modules, and of affine algebraic varieties over R are undecidable.

Isomorphisms of function modules, and generalized approximation in modulus

For a function algebra $A$ we investigate relations between the following three topics: isomorphisms of singly generated $A$-modules, Morita equivalence bimodules, and “real harmonic functions” with respect to $A$. We also consider certain groups which are naturally associated with a uniform algebra $A$. We illustrate the notions considered with several examples.

A cyclic six-term exact sequence for block matrices over a PID

Let be a 2×2 upper triangular block matrix over a principal ideal domain D with square diagonal blocks A 1 and A 2. We define a cyclic six-term exact sequence ϵ(A) in terms of the kernels and the cokernels of A 1 and A 2 with a connecting map defined by the off-diagonal block X. This cyclic sequence ϵ(A), under a variety of block-preserving matrix equivalences, is an invariant strictly finer than the Smith normal forms of A 1 A 2 and A combined. As one example of how this new K-theoretic invariant is used in classical linear algebra, we prove that 2×2 upper triangular block matrices A and B over a field F are block-preserving similar if and only if , that is, there is a chain F[t]-module isomorphism between the two cyclic sequences. We conclude with an application ϵ(A) to the flow equivalence classification of two-component shifts of finite type in symbolic dynamics. This paper is self-contained and presented in a matrix-theoretic form.

Deformation quantization and invariant distributions

In 1979, M. Kashiwara and M. Vergne formulated a conjecture on a Lie group G which implies that the Duflo isomorphism of Z(g) and S(g)^g extends to a natural module isomorphism between the spaces of germs of invariant distributions on G and g=Lie(G), respectively. They also proved their conjecture for G solvable. Using Kontsevich's deformation quantization we establish directly this result for distributions on any real Lie group G. In turn this gives a new proof of Duflo's result on the local solvability of bi-invariant differential operators on G.

G-fonctions et cohomologie des hypersurfaces singulières

Our object of study is the arithmetic of the differential modules (l) (l ∈ ℕ – {0}), associated by Dwork's theory to a homogeneous polynomial f (λ,X) with coefficients in a number field. Our main result is that (1) is a differential module of type G, c'est-à-dire, a module those solutions are G-functions. For the proof we distinguish two cases: the regular one and the non regular one.Our method gives us an effective upper bound for the global radius of (l), which doesn't depend on “l” but only on the polynomial f (λ,X). This upper bound is interesting because it gives an explicit estimate for the coefficients of the solutions of (l).In the regular case we know there is an isomorphism of differential modules between (1) and a certain De Rham cohomology group, endowed with the Gauss-Manin connection, c'est-à-dire, our module “comes from geometry”. Therefore our main result is a particular case of André's theorem which assert that at least in the regular case, all modules coming from geometry are of type G.